TC 310 MODES OF REASONING D. Hamermesh Fall 2002 Answer Key, First Midterm, September 26, 2002 1. (12 minutes) Jane wants to meet someone new and is looking through the Austin Chronicle personal ads. She is picky about who she goes out with, and is particularly concerned about height. Her ideal date is someone who is within one inch of her height, which is 67’’ (i.e., she wants someone between 66 and 68 inches). Here are the heights of men advertising in the Chronicle . Height: 71 73 72 70 72 69 70 68 68 69 71 70 67 a. Calculate the mean and standard deviation for this sample of men. Calculate the coefficient of variation. Calculate the median too. Mean = (sum of observations)/(number of observations) = 980/13 = 70 St. Dev= RMS of the deviations from the mean = 1.71 or 1.78 CV = sd x /mean of x = 1.71/70 = .024 or 1.78/70=.025 Median: 70 b. Draw a histogram of the information. Can you convey the information in a way that would be especially useful to Jane? Does the density function appear to be well approximated by the normal? The area under the histogram should sum to 100%. 0. 0 05 . 0 1 . 0 15 . 0 2 . 0 25 The distribution looks normal. It is symmetric around the mean. However, there are so few observations that even one more at the ends would skew the distribution. 2. (12 minutes) a. Many of the most intriguing ads don’t provide a height. What are Jane’s chances of meeting an ideal date if she responds to an ad without a reported height? Use the mean and standard deviation above and assume the distribution is normal. Z1 [Pr(HEIGHT)<68] = (68-70)/1.71 = -.2594 Pr(x<Z1) = .1211 Z2 [Pr(HEIGHT)<66] = (66-70)/1.71 = -2.3392 Pr(x<Z2) = .0097 .1211 - .0097 = .1114 or 11.14%

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b. What crucial assumption did you need to make about the heights of men who don’t report their heights compared with those who do? Is this assumption realistic? How does this influence your estimate of Jane’s probability of getting an ideal date? You need to assume that the heights of those who don’t report are distributed the same (with the same mean and standard deviation) as those who report their height – i.e. that the reported heights are a true random sample of men who put ads in. If men assume that being taller is preferable to being short, then the mean of height of the reported sample may be higher than the mean of height for the entire population of men placing ads. It’s difficult to tell how this influences Jane’s likelihood of getting her ideal height. Since her ideal is below the mean, if the population mean were somewhat less she would have more chance of getting lucky. If the mean was much lower than her preferences, however, she might even have less chance. If men thought being short was preferable, than the height reported would underestimate true height, and Jane would have less probability of an ideal date. Also, the standard deviation of the sample is likely to be greater than for the self-reported

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